Method and apparatus for automated regularization tuning in magnetic resonance imaging (MRI) using compressed sensing (CS)

ABSTRACT

An apparatus and method are provided to simultaneously provide good image quality and fast image reconstruction from magnetic resonance imaging (MRI) data by selecting an appropriate value for the regularization parameter used in compressed sensing (CS) image reconstruction. In CS reconstruction a high-resolution image can be reconstructed from randomized undersampled data by imposing sparsity in multi-scale transformation (e.g., wavelet) domain. Further, in the transformation domain, a threshold can be determined between signal and noise levels of the transform coefficients. A regularization parameter based on this threshold scales the regularization term, which imposes sparsity, relative to the data fidelity term in an objective function, thereby balancing the tradeoff between noise and smoothing.

FIELD

This disclosure relates to magnetic resonance imaging (MRI) usingcompressed sensing (CS), and, more particularly, to selecting an amountof regularization in the CS reconstruction method to provide good imagequality.

BACKGROUND

The background description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventors, to the extent the work is described in thisbackground section, as well as aspects of the description that cannototherwise qualify as prior art at the time of filing, are neitherexpressly nor impliedly admitted as prior art against the presentdisclosure.

Medical imaging produces images of the internal members of a patient'sbody. For example, magnetic resonance imaging (MRI) uses radio waves,magnetic fields, and magnetic-field gradients to produce images of theinternal members of a patient's body. Medical-imaging modalities alsoinclude, for example, X-ray radiography, ultrasonography, computedtomography (CT), and positron emission tomography (PET). Once the imageshave been produced, a physician can use the images to diagnose apatient's injuries or diseases.

Also, some medical-imaging devices use compressed sensing (CS). CSreconstructs images using a lower sampling rate than the rate requiredby the Nyquist-Shannon sampling theorem. CS takes advantage of animage's sparsity within a given domain. Some examples of a domain inwhich an image may be sparse include the spatial domain (conventionalthree-dimensional space), time domain (for a time series of images), orwavelet domain (data produced via wavelet transform). CS is able torecover the image using the lower sampling rate using iterativereconstruction methods, which are typically slow and computationallyexpensive.

The image quality obtained during MRI performed using CS can depend onthe degree of regularization. In general, more regularization leads togreater smoothing and less noise, but too much smoothing can lead toblurring and reduced resolution.

In certain implementations of CS, the degree of regularization can becontrolled by a regularization parameter β. For example, CS can beperformed by minimizing an objective function that includes a datafidelity term and a regularization term. The regularization term can bescaled by the regularization parameter β, which multiplies theregularization term, and an optimal value for the regularizationparameter β will balance the tradeoffs between noise and resolution.

Accordingly, improved methods are desired to select the value of theregularization parameter β that is to be applied during CS MRI imagereconstruction to optimize image quality.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of this disclosure is provided byreference to the following detailed description when considered inconnection with the accompanying drawings, wherein:

FIG. 1 illustrates a schematic block diagram of an magnetic resonanceimaging (MRI) system, according to one implementation;

FIG. 2 illustrates a flow diagram of a method for determining aregularization parameter and applying it to compressed sensing (CS)image reconstruction, according to one implementation;

FIG. 3A illustrates a reconstructed image of a head phantom, generatedusing a regularization parameter selected by a user, according to oneimplementation;

FIG. 3B illustrates a reconstructed image of the head phantom, generatedusing a regularization parameter selected by an automated methoddescribed herein, according to one implementation;

FIG. 4A illustrates a reconstructed image of a chest, generated using aregularization parameter selected by a user, according to oneimplementation;

FIG. 4B illustrates a reconstructed image of the chest, generated usinga regularization parameter selected by the automated method describedherein, according to one implementation; and

FIG. 5 illustrates a more detailed schematic block diagram of the MRIsystem.

DETAILED DESCRIPTION

Exemplary embodiments are illustrated in the referenced figures of thedrawings. It is intended that the embodiments and figures disclosedherein are to be considered illustrative rather than restrictive. Nolimitation on the scope of the technology and of the claims that followis to be imputed to the examples shown in the drawings and discussedherein.

The embodiments are mainly described in terms of particular processesand systems provided in particular implementations. However, theprocesses and systems will operate effectively in other implementations.Phrases such as ‘an embodiment’, ‘one embodiment’, and ‘anotherembodiment’ can refer to the same or different embodiments. Theembodiments will be described with respect to methods and compositionshaving certain components. However, the methods and compositions caninclude more or less components than those shown, and variations in thearrangement and type of the components can be made without departingfrom the scope of the present disclosure.

The exemplary embodiments are described in the context of methods havingcertain steps. However, the methods and compositions operate effectivelywith additional steps and steps in different orders that are notinconsistent with the exemplary embodiments. Thus, the presentdisclosure is not intended to be limited to the embodiments shown, butis to be accorded the widest scope consistent with the principles andfeatures described herein and as limited only by the appended claims.

Furthermore, where a range of values is provided, it is to be understoodthat each intervening value between an upper and lower limit of therange—and any other stated or intervening value in that stated range—isencompassed within the disclosure. Where the stated range includes upperand lower limits, ranges excluding either of those limits are alsoincluded. Unless expressly stated, the terms used herein are intended tohave the plain and ordinary meaning as understood by those of ordinaryskill in the art. Any definitions are intended to aid the reader inunderstanding the present disclosure, but are not intended to vary orotherwise limit the meaning of such terms unless specifically indicated.

As discussed above, improved methods are desired for selecting anoptimal value of the regularization parameter that is to be appliedduring CS magnetic resonance imaging (MRI) image reconstruction.

In certain implementations of CS image reconstruction, the degree ofregularization is controlled by a regularization parameter β in anobjective function. For example, CS reconstruction can be performed byfinding the reconstructed image x that minimize an objective function,in which the objective function includes a data fidelity term and aregularization term. Further, the regularization term can express asparsity condition that favors solutions in which a wavelet (or other)transformation of the reconstructed image x is sparse.

In one non-limiting example, CS reconstruction is performed by solvingthe optimization problem

${\overset{\hat{}}{u} = {{\arg{\min\limits_{u}{{y_{R} - {A_{R}{CW}^{\prime}u}}}_{2}^{2}}} + {\beta{u}_{1}}}},$wherein W is a wavelet transformation and W′ is an inverse wavelettransformation, û and u are the wavelet transformations of thereconstructed image x (i.e., û=W{circumflex over (x)} and {circumflexover (x)}=W′û), C is a matrix representing the spatial distribution ofthe receiver coil sensitivities, y_(R) is the acquired k-space data forall coils with variable density random undersampling of phase encoding(PE) lines, and A_(R) is a Fourier encoding matrix corresponding to thesampling pattern of the acquisition of y_(R), such that the HessianA′_(R)A_(R) does not have special structure. Here, the sparsitycondition is found in the regularization term ∥u∥₁, and the sparsitycondition is expressed by applying the l₁ norm (i.e., ∥⋅∥₁) to thewavelet transformation u of the reconstructed image x. In the objectivefunction, the expression ∥y_(R)−A_(R)CW′u∥₂ ² is the data fidelity term,which is minimized in order to approximately solve the matrix equationAx=y_(R), wherein A=A_(R)C and x=W′u. When the regularization parameterβ is increased sparser solutions are favored at the expense of lowerdata fidelity, and, for a smaller regularization parameter β, improveddata fidelity is encouraged at the expense of decreased sparsity. Inthis model, the regularization parameter β balances the relativecontributions to the objective functions between data fidelity term andthe regularization term.

To select an optimal regularization parameter β, an iterative methodbased on the discrepancy principle was suggested in K. F. King et al.“Adaptive regularization in compressed sensing using the discrepancyprinciple,” Proc. Intl. Soc. Mag. Reson. Med. Vol 17, p. 2822 (2009),which is incorporated herein by reference in its entirety.Unfortunately, the discrepancy-principle based approach suffers frombeing computational intensive. For example, in each of the iterativeloops of the discrepancy-principle based approach, CS reconstruction isperformed using a given value for the regularization parameter β, thenthe regularization parameter β is updated based on the reconstructedimage and the loop is repeated until the residual of the data fidelityterm falls below a desired noise threshold. This can be slow because theCS reconstruction performed in each loop requires significantcomputations. In contrast, the methods described herein estimate thebest regularization parameter β before performing CS reconstruction,such that CS reconstruction is only performed once, resulting insignificant computational savings.

Referring now to the drawings, wherein like reference numerals designateidentical or corresponding parts throughout the several views, FIG. 1illustrates an example embodiment of a medical-imaging system 10. Themedical-imaging system 10 includes at least one scanning device 100; oneor more image-generation devices 110, each of which is aspecially-configured computing device (e.g., a specially-configureddesktop computer, a specially-configured laptop computer, aspecially-configured server); and a display device 120.

The scanning device 100 is configured to acquire scan data by scanning aregion (e.g., area, volume, slice) of an object (e.g., a patient). Thescanning modality may be, for example, magnetic resonance imaging (MRI),computed tomography (CT), positron emission tomography (PET), X-rayradiography, and ultrasonography. The scanning device 100 may acquire arandomly undersampled set of scan data that is appropriate for CSreconstruction or acquire scan data that may be used by a CS process.Accordingly, CS data includes scan data that may be used by a CS processor that is appropriate for CS reconstruction.

The one or more image-generation devices 110 obtain scan data from thescanning device 100 and generate an image of the region of the objectbased on the scan data. To generate the image, for example when the scandata is CS data, the one or more image-generation devices 100 mayperform a reconstruction process on the scan data. Examples ofreconstruction processes include GRAPPA, SENSE, ARC, SPIRiT, LORAKS,ISTA, and FISTA. For CS data, the reconstruction process can be anon-linear process that enforces both the sparsity of the imagerepresentation within a given domain (e.g., spatial, time, wavelet) andthe consistency of the reconstruction with the acquired scan data.

After the one or more image-generation devices 110 generate the image,the one or more image-generation devices 110 send the image to thedisplay device 120, which displays the image.

Also, the one or more image-generation devices 110 may generate twoimages from the same scan data. The one or more image-generation devices110 may use different reconstruction processes to generate the twoimages from the same scan data, and one image may have a lowerresolution than the other image. Additionally, the one or moreimage-generation devices 110 may generate an image.

FIG. 2 illustrates a non-limiting example of a flow diagram for using CSto reconstruct an MRI image. In addition to this non-limiting example,other flow diagrams may be used. For example, some embodiments mayperform at least some of the operations in different orders than thepresented orders. Examples of different orders include concurrent,parallel, overlapping, reordered, simultaneous, incremental, andinterleaved orders. Thus, other embodiments of the operational flowsthat are described herein may omit blocks, add blocks, change the orderof the blocks, combine blocks, or divide blocks into more blocks.

Furthermore, although this flow diagram is described as being performedby an image-generation device, some embodiments of the flow diagram canbe performed by two or more image-generation devices or by one or moreother specially-configured computing devices.

In step 210 of method 200, the image-generation device obtains scandata, which are defined in an acquisition space. For example, if thescan modality is MRI, then the acquisition space may be k-space, and thek-space data may be acquired using collection methods such as Cartesiansampling, spiral sampling, and radial sampling. Here, method 200 isillustrated for a case of MRI data sampled for CS, as would beunderstood by a person of ordinary skill in the art.

In step 210 of method 200, a preliminary image is generated. Forexample, the MRI data from step 210 can be acquired in k spaces throughrandomized undersampling. Then, the preliminary image can be generatedby zero filing the under-sampled MRI data and back-projecting to theimage domain to generate a preliminary image. That is, the sampled datay_(R) can be zero-filled to generate zero-filled data {tilde over(y)}_(R) in k-space, then the zero-filled data {tilde over (y)}_(R) canbe back-projected x₀=A′{tilde over (y)}_(R) to generate the preliminaryimage x₀, wherein A′ is the back-projection operator, which is theadjoint to the forward-projection operator A.

This preliminary image x₀ can also be used to provide a warm start foriterative reconstruction using a CS method. CS reconstruction can beperformed as an iterative method that searches for the reconstructedimage x that minimizes an objective function. This search starts with aninitial guess for the reconstructed image x, and a warm start (i.e., thepreliminary image) can shorten the search by providing an initial guessfor the search that is at least an informed approximation of the finalreconstructed image x. Other methods, besides back-projecting thezero-filled MRI data, can also be used to generate a preliminary imagex₀.

In certain implementations, the preliminary image x₀ that is used fordetermining the regularization parameter may or may not be the same asthe initial guess for iterative reconstruction using the CSreconstruction method.

In step 230 of method 200, a multi-scale transformation is performed onthe preliminary image x₀. To illustrate method 200, the wavelettransformation is used as one example of a multi-scale transformation.However, other multi-scale transformations can be used without departingfrom the spirit of the invention. Examples, of other multi-scaletransformations that can be used include: Haar transformations, Gabortransformations, curvelet transformations, Gaussian pyramidtransformations, Laplacian pyramid transformations, steerable pyramidtransformations, blocked discrete cosine transformations, and blockeddiscrete Fourier transformations.

Continuing with the non-limiting example that step 230 is performedusing a wavelet transformation, the preliminary image x₀ is transformedinto wavelet coefficients corresponding to different subbands (e.g., alow-low subband, a low-high subband, a high-low subband, and a high-highsubband). In this example, the high-high subband is a fine-scalesubband, and the low-low subband is a coarse-scale subband. The low-highsubband and the high-low subband would each represent features that arelarge/coarse in one spatial dimension and small/fine in another spatialdimension.

In step 240 of method 200, a thresholding method is applied to thewavelet coefficients of one of the subbands (e.g., the finest-scalesubband) to determine a threshold value. The threshold value is thenused in turn to calculate regularization parameter to be used in CSreconstruction.

In one example of an implementation of step 240, a threshold value isselected based on the transform coefficients of the finest scale subbandof the multi-scale transformation. This threshold value can be selected,e.g., using a histogram of the wavelet coefficients in the finest-scalesubband of the wavelet transformation, and selecting the coefficientvalue of the Nth percentile of the histogram. In certainimplementations, the Nth percentile of the histogram can be the 99thpercentile of the histogram. The Nth percentile of the histogram can besuch that the number of coefficients in the subband having values belowthe threshold is one order of magnitude greater than the number ofcoefficients in the subband having values greater than the threshold. Inother implementations, the Nth percentile of the histogram can be suchthat the number of coefficients in the subband having values below thethreshold is two orders of magnitude greater than the number ofcoefficients in the subband having values greater than the threshold.

For example, 10-20% of the coefficients in the subband can have valuesgreater than the threshold. In another implementation, 5-10% of thecoefficients in the subband can have values greater than the threshold.In a third implementation, 2.5-5% of the coefficients in the subband canhave values greater than the threshold. In a fourth implementation,1-2.5% of the coefficients in the subband can have values greater thanthe threshold.

In certain implementations, the regularization parameter is equal to thethreshold value. In other implementations, the regularization parameteris calculated using a function which includes the threshold value as oneof its inputs.

Although, the determination of the threshold value is illustrated usingthe example of a histogram based method, other methods of determiningthe threshold value can be used without departing from the spirit of theinvention. For example, the threshold value can be determined using anOtsu's method.

Otsu's method is used to automatically perform clustering-based imagethresholding. In Otsu's method, the algorithm assumes that the imagecontains two classes of pixels following bi-modal histogram (foregroundpixels and background pixels). The optimum threshold separating the twoclasses is then calculated so that their combined spread (intra-classvariance) is minimal. That is, the optimum threshold separating the twoclasses is then calculated so that their inter-class variance is maximalor equivalently (i.e., the sum of pairwise squared distances isconstant). Otsu's method can be better understood by noting that it isroughly a one-dimensional, discrete analog of Fisher's DiscriminantAnalysis, and that Otsu's method is also directly related to the Jenksoptimization method.

Further, in certain implementations, the threshold value can bedetermined using any one of a clustering-based thresholding method, ak-means clustering method, and/or a mixture-model based method.

In step 250 of method 200, CS reconstruction is performed on the MRIdata using an objective function in which the relative contributionsbetween the data fidelity term and the regularization term is based onthe threshold determined in step 240. In general, the imagereconstruction method can be any regularizer method (i.e., an imagereconstruction method that minimizes an objective function that includesa regularizer/regularization term), and is not limited to CS methods.

For example, in the case that the image x=W′u is reconstructed byminimizing the above objective function∥y_(R)−A_(R)CW′u∥₂ ²+β∥u∥₁,the regularization parameter β can be set equal to the threshold valueitself, which is the value of Nth percentile of a histogram of thefinest-scale subband of a wavelet transformation. More generally, theregularization parameter β can be a function of the threshold value.

The threshold value estimates a division between noise and signal valuesin the wavelet coefficients. The assumption of CS is that, in thetransform/wavelet domain, the transform/wavelet coefficients expressingsignal will be sparse, whereas most wavelet coefficients represent noisepredominantly. This sparsity assumption can be even stronger in thefiner-scale subbands than in the coarser-scale subbands, especially whenmost of the signal is expressed at lower spatial frequencies. Thus, thethreshold value determined in step 240 can estimate a cutoff demarkingthe boundary between coefficient values that represent signal and thosethat represent noise. This estimated threshold provides an appropriatescaling factor to relate the regularization term to the data fidelityterm of the objective function. Further, by selecting the thresholdbased on the wavelet transformation of the MRI data itself, thedetermination of the regularization parameter is robust againstvariations in the patient's size, factors affecting the signal and noiselevels, acceleration factors, coil choice, the MRI scanner setup,geometry, and orientation.

The choice of N in the Nth percentile of a histogram can be selectedbased on empirical factors and observations in order to obtain preferredimage characteristics and quality. Advantageously, the empirical tuningonly needs to happen once (e.g., in the factory)—not on a per protocolbasis.

In addition to the non-limiting example of method 200 discussed above,variations of the method can be implemented without departing from thespirit of the invention. In the above-described implementation, aregularization parameter is based on the threshold from the finest-scalesubband, and this regularization parameter can be used for all subbands.

In other implementations, the regularization parameter can be based on athreshold that is determined using a different subband than thefinest-scale subband, and this regularization parameter can be used forall subbands.

In still other implementations, multiple regularization parameters canbe used, and these different regularization parameters can be based onrespective thresholds estimated from various subbands of the multi-scaletransformation. Then, these different regularization parameters can beused for different regularization terms in the objective function. Forexample, objective function might include that the regularization termis split into two or more regularization terms. In certainimplementations, the objective function can include a fineregularization term and a coarse regularization term, i.e.,

${\overset{\hat{}}{u} = {{\arg{\min\limits_{u}{{y_{R} - {A_{R}{CW}^{\prime}u}}}_{2}^{2}}} + {\beta_{H}{u_{H}}_{1}} + {\beta_{L}{u_{L}}_{1}}}},$wherein u={u_(L), u_(H)} and u_(H) is one subset of the wavelet subbands(e.g., the fine subbands) and u_(L) is another subset of the rest of thewavelet subbands (e.g., the coarse subbands). Then the firstregularization parameter β_(H) can be based on a subband from the firstsubset u_(H) of the wavelet subbands. Further, the second regularizationparameter β_(L) can be based on a subband from the second subset u_(L)of the wavelet subbands.

More generally, the subbands can be partitioned into any number L ofsubsets, and the objective function can be expressed as

${\overset{\hat{}}{u} = {{\arg{\min\limits_{u}{{y_{R} - {A_{R}{CW}^{\prime}u}}}_{2}^{2}}} + {\sum\limits_{i = 1}^{L}\;{\beta_{i}{u_{i}}_{1}}}}},$

wherein u={u₁, u₂, . . . , u_(L)}. For example, the number L of subsetscan be equal to the number of subbands. Then, for each subband, thethreshold and regularization parameter would be based on a thresholdingmethod performed on the transform coefficients of the respectivesubband.

In still other implementations, threshold values from multiple subbandscan be averaged/combined to generate a single regularization parameter.

Alternatively, the noise can be extracted/estimated in k-space, and thenback-projected into the image domain before performing the multi-scaletransformation to generate respective subbands. In this case, eachsubband would represent isolated noise, without any signal. Thethreshold could then be estimated based on the noise alone. For example,the noise can be estimated/measured by performing calibration scanwithout exciting the nuclear spins.

FIGS. 3A and 3B show reconstructed images from MRI data from a headphantom. FIG. 3A shows the reconstructed image when the regularizationparameter, which was selected by a user was too high (i.e., 2.0),resulting in the regularization/sparsity condition being overemphasizedrelative to the data fidelity condition. This overemphasis on sparsityresults in the periodic, vertical striation artifact observable in theneck region. FIG. 3B shows the reconstructed image when theregularization parameter is determined using the method describedherein. In FIG. 3B, the regularization parameter is set to 0.8 ratherthan 2.0. By decreasing the regularization parameter to a more optimalvalue, the striation artifact is mitigated.

FIGS. 4A and 4B show reconstructed images from MRI data of a chest.Here, like in FIG. 3A, FIG. 4A shows an image in which the user selectedby a user regularization parameter was too high (i.e., 2.8), resultingin too much smoothing, although there are no obvious artifacts. FIG. 4Bshows the reconstructed image when the regularization parameter isdetermined using the method described herein. In FIG. 4B, theregularization parameter is set to 1.36 rather than 2.8. By decreasingthe regularization parameter to a more optimal value, the structurewithin the soft tissue and the lungs becomes more visible.

In summary, the methods described herein for automatically determiningthe regularization parameter have several advantages. First, the methodsdescribed herein simplify workflow and decrease the opportunities forerror by eliminating the requirement for human interaction to select theregularization parameter. Further, the methods described herein provideoptimization of the image quality by balancing the trade-offs betweennoise and smoothing/blurring. Third, the methods described herein arerobust to changes in protocols, patient size, accelerations factors,noise/signal levels, etc., thereby providing image quality consistencyfor all protocols. Fourth, the methods described herein are fast becausethey can be performed with a preliminary image, and do not require aniterative process in which each loop includes CS image reconstruction.Accordingly, the methods described herein add little additionalcomputation time to existing CS reconstruction methods. This achievessignificant computation and time savings relative todiscrepancy-principle based methods, which require multiple, sequentialCS reconstructions.

FIG. 5 shows a non-limiting example of a magnetic resonance imaging(MRI) system 10. The MRI system 10 depicted in FIG. 5 includes a gantry501 (shown in a schematic cross-section) and various related systemcomponents 503 interfaced therewith. At least the gantry 501 istypically located in a shielded room. The MRI system geometry depictedin FIG. 5 includes a substantially coaxial cylindrical arrangement ofthe static field B₀ magnet 511, a Gx, Gy, and Gz gradient coil set 513,and a large whole-body RF coil (WBC) assembly 515. Along a horizontalaxis of this cylindrical array of elements is an imaging volume 517shown as substantially encompassing the head of a patient 519 supportedby a patient table 520.

One or more smaller array RF coils 521 can be more closely coupled tothe patient's head (referred to herein, for example, as “scanned object”or “object”) in imaging volume 517. As those in the art will appreciate,compared to the WBC (whole-body coil), relatively small coils and/orarrays, such as surface coils or the like, are often customized forparticular body parts (e.g., arms, shoulders, elbows, wrists, knees,legs, chest, spine, etc.). Such smaller RF coils are referred to hereinas array coils (AC) or phased-array coils (PAC). These can include atleast one coil configured to transmit RF signals into the imagingvolume, and a plurality of receiver coils configured to receive RFsignals from an object, such as the patient's head, in the imagingvolume.

The MRI system 10 includes a MRI system controller 530 that hasinput/output ports connected to a display 524, a keyboard 526, and aprinter 528. As will be appreciated, the display 524 can be of thetouch-screen variety so that it provides control inputs as well. A mouseor other I/O device(s) can also be provided.

The MRI system controller 530 interfaces with a MRI sequence controller540, which, in turn, controls the Gx, Gy, and Gz gradient coil drivers532, as well as the RF transmitter 534, and the transmit/receive switch536 (if the same RF coil is used for both transmission and reception).The MRI sequence controller 540 includes suitable program code structure538 for implementing MRI imaging (also known as nuclear magneticresonance, or NMR, imaging) techniques including parallel imaging. MRIsequence controller 540 can be configured for MR imaging with or withoutparallel imaging. Moreover, the MRI sequence controller 540 canfacilitate one or more preparation scan (pre-scan) sequences, and a scansequence to obtain a main scan magnetic resonance (MR) image (referredto as a diagnostic image). MR data from pre-scans can be used, forexample, to determine sensitivity maps for RF coils 515 and/or 521(sometimes referred to as coil sensitivity maps or spatial sensitivitymaps), and to determine unfolding maps for parallel imaging.

The MRI system components 503 include an RF receiver 541 providing inputto data processor 542 so as to create processed image data, which issent to display 524. The MRI data processor 542 is also configured toaccess previously generated MR data, images, and/or maps, such as, forexample, coil sensitivity maps, parallel image unfolding maps,distortion maps and/or system configuration parameters 546, and MRIimage reconstruction program code structures 544 and 550.

In one embodiment, the MRI data processor 542 includes processingcircuitry. The processing circuitry can include devices such as anapplication-specific integrated circuit (ASIC), configurable logicdevices (e.g., simple programmable logic devices (SPLDs), complexprogrammable logic devices (CPLDs), and field programmable gate arrays(FPGAs), and other circuit components that are arranged to perform thefunctions recited in the present disclosure.

The processor 542 executes one or more sequences of one or moreinstructions contained in the program code structures 544 and 550 (e.g.,method 200). Alternatively, the instructions can be read from anothercomputer-readable medium, such as a hard disk or a removable mediadrive. One or more processors in a multi-processing arrangement can alsobe employed to execute the sequences of instructions contained in theprogram code structures 544 and 550 (e.g., method 200). In alternativeembodiments, hard-wired circuitry can be used in place of or incombination with software instructions. Thus, the disclosed embodimentsare not limited to any specific combination of hardware circuitry andsoftware.

Additionally, the term “computer-readable medium” as used herein refersto any non-transitory medium that participates in providing instructionsto the processor 542 for execution. A computer readable medium can takemany forms, including, but not limited to, non-volatile media orvolatile media. Non-volatile media includes, for example, optical,magnetic disks, and magneto-optical disks, or a removable media drive.Volatile media includes dynamic memory.

Also illustrated in FIG. 5 is a generalized depiction of an MRI systemprogram storage (memory) 550, where stored program code structures arestored in non-transitory computer-readable storage media accessible tothe various data processing components of the MRI system 10. As those inthe art will appreciate, the program store 550 can be segmented anddirectly connected, at least in part, to different ones of the system503 processing computers having most immediate need for such storedprogram code structures in their normal operation (i.e., rather thanbeing commonly stored and connected directly to the MRI systemcontroller 530).

While certain implementations have been described, these implementationshave been presented by way of example only, and are not intended tolimit the teachings of this disclosure. Indeed, the novel methods,apparatuses and systems described herein can be embodied in a variety ofother forms; furthermore, various omissions, substitutions and changesin the form of the methods, apparatuses and systems described herein canbe made without departing from the spirit of this disclosure.

The invention claimed is:
 1. An imaging apparatus, comprising:processing circuitry configured to obtain magnetic resonance imaging(MRI) data, the MRI data being acquired by non-uniformly samplingk-space, perform a multi-scale transformation on a preliminary imagegenerated from the MRI data and generating thereby a plurality ofsubbands, each of the plurality of subbands comprising respectivetransformation coefficients of the multi-scale transformation, whereinthe multi-scale transformation is one of a curvelet transformation, aGaussian pyramid, a Laplacian pyramid, a steerable pyramid, a Gabortransformation, a blocked discrete cosine transformation and a blockeddiscrete Fourier transformation, determine, based on at least one of theplurality of subbands, a threshold between signal and noise of thetransformation coefficients, and reconstruct an MRI image from the MRIdata using a regularized method in which an objective function isminimized, the objective function including a data fidelity term and aregularization term that is scaled based on the determined threshold. 2.The apparatus according to claim 1, wherein the processing circuitry isfurther configured to determine the threshold using one or more of aclustering-based thresholding method, an Otsu's method, a k-meansclustering method, a mixture-model based method, and/or a method thatincludes setting the threshold to a value of a predetermined percentileof a histogram of the transformation coefficients of the one of theplurality of subbands.
 3. The apparatus according to claim 1, whereinthe processing circuitry is further configured to reconstruct the MRIimage from the MRI data using respective thresholds corresponding torespective subbands of the plurality of subbands, each of the respectivethresholds scaling a respective regularization term of the objectivefunction corresponding to a respective subband.
 4. The apparatusaccording to claim 1, wherein the processing circuitry is furtherconfigured to reconstruct the MRI image from the MRI data by minimizingthe objective function, wherein the regularization term of the objectivefunction imposes sparsity on the MRI image in a domain of themulti-scale transformation.
 5. The apparatus according to claim 1,wherein the processing circuitry is further configured to determine thethreshold using a calibration measurement representing a noise level ofthe MRI data.
 6. An imaging apparatus comprising: processing circuitryconfigured to: obtain magnetic resonance imaging (MRI) data, the MRIdata being acquired by non-uniformly sampling k-space, generate apreliminary image from the MRI data by back-projecting a zero-filledFourier transform of the MRI data, perform a multi-scale transformationon the preliminary image thereby generating a plurality of subbands,wherein the multi-scale transformation is a wavelet transformation andeach of the plurality of subbands comprises respective transformationcoefficients of the multi-scale transformation, determine, based on atleast one of the plurality of subbands, a threshold between signal andnoise of the transformation coefficients based on a histogram of afinest-scale subband of the wavelet transformation of the preliminaryimage, and reconstruct an MRI image from the MRI data using aregularized method in which an objective function is minimized, theobjective function including a data fidelity term and a regularizationterm that is scaled based on the determined threshold.
 7. The apparatusaccording to claim 6, wherein the processing circuitry is furtherconfigured to reconstruct the MRI image from the MRI data by minimizingthe objective function, wherein the regularization term of the objectivefunction is an l₁-norm of the wavelet transformation of the MRI image,and the scaling of the regularization term is performed by multiplyingthe regularization term by a regularization parameter that isproportional to a value of a predetermined percentile of the histogramof the finest-scale subband of the wavelet transformation of thepreliminary image.
 8. The apparatus according to claim 6, furthercomprising a radio frequency (RF) transmitter configured to radiate asequence of RF pulses to generate magnetic resonance (MR) signals, amagnetic field coil configured to generate a sequence of gradientmagnetic fields corresponding to an encoding pattern that non-uniformlysamples k-space, and an RF receiver configured to receive MR signals andgenerate the MR data.
 9. The apparatus according to claim 6, wherein theprocessing circuitry is further configured to reconstruct the MRI imagefrom the MRI data using respective thresholds corresponding torespective subbands of the plurality of subbands, each of the respectivethresholds scaling a respective regularization term of the objectivefunction corresponding to a respective subband.
 10. The apparatusaccording to claim 6, wherein the processing circuitry is furtherconfigured to reconstruct the MRI image from the MRI data by minimizingthe objective function, wherein the regularization term of the objectivefunction imposes sparsity on the MRI image in a domain of themulti-scale transformation.
 11. An imaging apparatus comprising:processing circuitry configured to obtain magnetic resonance imaging(MRI) data, the MRI data being acquired by non-uniformly samplingk-space, perform a multi-scale transformation on a preliminary imagegenerated from the MRI data and generating thereby a plurality ofsubbands, each of the plurality of subbands comprising respectivetransformation coefficients of the multi-scale transformation,determine, based on at least one of the plurality of subbands, athreshold between signal and noise of the transformation coefficients,and reconstruct an MRI image from the MRI data using a regularizedmethod in which an objective function is minimized, the objectivefunction including a data fidelity term and a regularization term thatis scaled based on the determined threshold, wherein the thresholdrepresents a cutoff between transformation coefficients having largevalues representing MRI signals and transformation coefficients havingsmall values representing noise, a number of transformation coefficientsbelow the cutoff being an order of magnitude greater than a number oftransformation coefficients above the cutoff.
 12. An imaging apparatuscomprising: processing circuitry configured to obtain magnetic resonanceimaging (MRI) data, the MRI data being acquired by non-uniformlysampling k-space, perform a multi-scale transformation on a preliminaryimage generated from the MRI data and generating thereby a plurality ofsubbands, each of the plurality of subbands comprising respectivetransformation coefficients of the multi-scale transformation,determine, based on a first subband of the plurality of subbands and asecond subband of the plurality of subbands, a threshold between signaland noise of the transformation coefficients, and reconstruct an MRIimage from the MRI data using a regularized method in which an objectivefunction is minimized, the objective function including a data fidelityterm and a regularization term that is scaled based on the determinedthreshold, wherein the first subband is used to determine a first cutoffvalue between signal and noise transformation coefficients within thefirst subband, the second subband is used to determine a second cutoffvalue between signal and noise transformation coefficients within thesecond subband, and the threshold is a combination of the first cutoffvalue and the second cutoff value.
 13. An imaging apparatus, comprising:processing circuitry configured to obtain magnetic resonance imaging(MRI) data, the MRI data being acquired by non-uniformly samplingk-space, perform a multi-scale transformation on a preliminary imagegenerated from the MRI data and generating thereby a plurality ofsubbands, each of the plurality of subbands comprising respectivetransformation coefficients of the multi-scale transformation,determine, based on a first subband of the plurality of subbands and asecond subband of the plurality of subbands, a threshold between signaland noise of the transformation coefficients, and reconstruct an MRIimage from the MRI data using a regularized method in which an objectivefunction is minimized, the objective function including a data fidelityterm and a regularization term that is scaled based on the determinedthreshold, wherein the first subband is used to determine a first cutoffvalue between signal and noise transformation coefficients within thefirst subband to generate a first threshold, the second subband is usedto determine a second cutoff value between signal and noisetransformation coefficients within the second subband to generate asecond threshold, and the objective function includes two regularizationterms including a first regularization term that is scaled based on thefirst threshold and a second regularization term that is scaled based onthe second threshold, each of the first and second regularization termsimposing sparsity on the transformation coefficients of the first andsecond subsets respectively.
 14. A method comprising: obtaining magneticresonance imaging (MRI) data, the MRI data being acquired bynon-uniformly sampling k-space; performing a multi-scale transformationon a preliminary image generated from the MRI data and generatingthereby a plurality of subbands, each of the plurality of subbandscomprising respective transformation coefficients of the multi-scaletransformation, wherein the multi-scale transformation is one of acurvelet transformation, a Gaussian pyramid, a Laplacian pyramid, asteerable pyramid, a Gabor transformation, a blocked discrete cosinetransformation and a blocked discrete Fourier transformation;determining, based on at least one of the plurality of subbands, athreshold between signal and noise of the transformation coefficients;and reconstructing an MRI image from the MRI data using a regularizedmethod in which an objective function is minimized, the objectivefunction including a data fidelity term and a regularization term thatis scaled based on the determined threshold.
 15. The method according toclaim 14, wherein determining the threshold is performed using one ormore of a clustering-based thresholding method, an Otsu's method, ak-means clustering method, a mixture-model based method, and/or a methodthat includes setting the threshold to a value of a predeterminedpercentile of a histogram of the transformation coefficients of the oneof the plurality of subbands.
 16. The method according to claim 14,wherein reconstructing the MRI image further includes that the objectivefunction is minimized wherein the regularization term imposes sparsityon the MRI image in a domain of the multi-scale transformation.
 17. Anon-transitory computer readable storage medium including executableinstructions, wherein the instructions, when executed by circuitry,cause the circuitry to perform the method according to claim
 14. 18. Amethod comprising: obtaining magnetic resonance imaging (MRI) data, theMRI data being acquired by non-uniformly sampling k-space; generating apreliminary image from the MRI data by back-projecting a zero-filledFourier transform of the MRI data; performing a multi-scaletransformation on the preliminary image thereby generating a pluralityof subbands, wherein the multi-scale transformation is a wavelettransformation and each of the plurality of subbands comprisesrespective transformation coefficients of the multi-scaletransformation; determining, based on at least one of the plurality ofsubbands, a threshold between signal and noise of the transformationcoefficients based on a histogram of a finest-scale subband of thewavelet transformation of the preliminary image; and reconstructing anMRI image from the MRI data using a regularized method in which anobjective function is minimized, the objective function including a datafidelity term and a regularization term that is scaled based on thedetermined threshold.
 19. The method according to claim 18, whereinreconstructing the MRI image is performed by minimizing the objectivefunction, wherein the regularization term of the objective function isan l₁-norm of the wavelet transformation of the MRI image, and thescaling of the regularization term is performed by multiplying theregularization term by a regularization parameter that is proportionalto a value of a predetermined percentile of the histogram of thefinest-scale subband of the wavelet transformation of the preliminaryimage.
 20. A method comprising: obtaining magnetic resonance imaging(MRI) data, the MRI data being acquired by non-uniformly samplingk-space; performing a multi-scale transformation on a preliminary imagegenerated from the MRI data and generating thereby a plurality ofsubbands, each of the plurality of subbands comprising respectivetransformation coefficients of the multi-scale transformation;determining, based on a first subband of the plurality of subbands and asecond subband of the plurality of subbands, a threshold between signaland noise of the transformation coefficients; and reconstructing an MRIimage from the MRI data using a regularized method in which an objectivefunction is minimized, the objective function including a data fidelityterm and a regularization term that is scaled based on the determinedthreshold; wherein the first subband is used to determine a first cutoffvalue between signal and noise transformation coefficients within thefirst subband, the second subband is used to determine a second cutoffvalue between signal and noise transformation coefficients within thesecond subband, and the threshold is a combination of the first cutoffvalue and the second cutoff value.